Historical note
(Numbers in brackets refer to the bibliography at the end of this note.)
We have already had occasion to remark that the development of analytic geometry of the plane and in space led mathematicians to the notion of -dimensional space, which provided them with an extremely convenient geometrical language for expressing simply and concisely algebraic theorems about equations in an arbitrary number of variables, and in particular all the general results of linear algebra. But although this language had become customary with many geometers by the middle of the 19th century, it remained purely a matter of convenience, and the absence of an "intuitive" representation of spaces of more than three dimensions appeared to forbid, in such spaces, the arguments "by continuity" which, founded exclusively on "intuition", were permitted in the plane and in (three-dimensional) space. In his researches on analysis situs and on the foundations of geometry, Riemann was the first to use considerations of this sort, by analogy with three-dimensional space (see the Historical Note on Chapter I) (*); following his example, many mathematicians were encouraged to use such arguments, with great success, particularly in the theory of algebraic functions of several complex variables. But in view of the very limited control over spatial intuition at that period, one might justifiably remain skeptical of the demonstrative value of such considerations, and not allow their use except on a purely heuristic basis, as tending to make plausible the truth of certain theorems. Thus Poincaré, in his memoir of 1887 on the residues of double integrals of two complex variables, avoided, as far as he could, all recourse to intuition in four-dimensional space: "Comme cette langue hypergéométrique répugne encore à beaucoup de bons esprits,
(*) See also the works of L. Schlafli, which date from the same period but remained unpublished until this century [3].
je n’en ferai qu’un usage peu fréquent"; the artifices which he used for this purpose enabled him to make topological arguments suffice when he was discussing three-dimensional space, he no longer hesitated to appeal for which to intuition ([1]).
Moreover, the discoveries of Cantor, and particularly the famous theorem which asserts that and are equipotent [which seemed to undermine the whole concept of dimension (*)], showed that, in order to put geometry and topology on a solid foundation, it was indispensable to free them entirely from all dependence on intuition. We have already noted (cf. Historical Note on Chapter I) that this need was the origin of the modern conception of general topology; but even before the creation of this latter theory there had already begun a rigorous study of the topology of real -dimensional spaces and their immediate generalizations ("-dimensional manifolds") by methods which belong properly to that branch of topology known as "combinatorial topology", or better "algebraic topology". A volume in this series will be devoted to this branch of topology, and the reader will find information there on the historical stages of its development; in this chapter we have limited ourselves to establishing the most elementary topological properties of real number spaces and real projective spaces which, historically, have served as the starting point for the methods of algebraic topology.
(*) It is interesting to note that Dedekind, as soon as he knew of this result, understood the reason for its paradoxical appearance, and remarked to Cantor that it ought to be possible to prove the impossibility of a bicontinuous one-to-one correspondence between and when [2].
[1] L. SCHLÄFLI, Gesammelte mathematische Abhandlungen, vol. 1, Basel (Birkhauser), 1950, pp. 169-387. [2] H. POINCARÉ, Œuvres, vol. 3, p. 443 et seq. Paris (Gauthier-Villars) (1934) [Acta Mathematica, 9 (1887), p. 321]. [3] G. CANTOR and R. DEDEKIND. Briefwechsel, Act. Sci. et Ind., no. 518, Paris (Hermann) (1937).